Geometric, Algebraic and Topological Methods for Quantum Field Theory

378 Pages·2013·2.57 MB·English

by  Alexander Cardona (ed.)

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Proceedings of the 2011 Villa de Leyva Summer School
Geometric, Algebraic
and
Topological Methods
Quantum Field Theory
for
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September28,2013 13:9 WSPC-ProceedingsTrimSize:9inx6in GATMQFT-VdL2011-PROCEEDINGS
Proceedings of the 2011 Villa de Leyva Summer School
Geometric, Algebraic
and
Topological Methods
Quantum Field Theory
for
Villa de Leyva, Colombia 4 – 22 July 2011
Editors
Alexander Cardona
Universidad de los Andes, Colombia
Carolina Neira-Jiménez
Universität Regensburg, Germany
Hernán Ocampo
Universidad del Valle, Colombia
Sylvie Paycha
Universität Potsdam, Germany
Andrés F Reyes-Lega
Universidad de los Andes, Colombia
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
September28,2013 13:9 WSPC-ProceedingsTrimSize:9inx6in GATMQFT-VdL2011-PROCEEDINGS
Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Villa de Leyva Summer School (7th : 2011 : Leiva, Boyacá, Colombia)
Geometric, algebraic and topological methods for quantum field theory : proceedings of the 2011
Villa de Leyva Summer School, Villa de Leyva, Colombia, 4–22 July 2011 / edited by Alexander
Cardona (Universidad de los Andes, Colombia), Carolina Neira-Jiménez (Universität Regensburg,
Germany), Hernán Ocampo (Universidad del Valle, Colombia), Sylvie Paycha (Universität Potsdam,
Germany) & Andrés F Reyes-Lega (Universidad de los Andes, Colombia).
pages cm
Includes bibliographical references and index.
ISBN 978-981-4460-04-0 (hardcover : alk. paper)
1. Geometric quantization--Congresses. 2. Quantum field theory--Mathematics--Congresses.
3. Topology--Congresses. I. Cardona, Alexander, editor of compilation. II. Title.
QC174.17.G46G455 2011
530.14'30151--dc23
2013037184
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Printed in Singapore
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v
INTRODUCTION
This volume offers an introduction to some recent developments in sev-
eral active topics at the interface between geometry, topology, analysis and
quantum field theory:
- Spectral geometry and index theory,
- Noncommutative models and the standard model,
- Algebraic geometry and Feynman periods,
- Generalized geometries and string theory,
- Differential geometry and gravity,
- Conformal field theory and integrability.
Itisbasedonlecturesandshortcommunicationsdeliveredduringasum-
merschoolon“Geometric,AlgebraicandTopologicalMethodsforQuantum
Field Theory” held in Villa de Leyva, Colombia, in July 2011. This school
wastheseventhofaseriesofsummerschoolswhichhavetakenplaceevery
other year since July 1999. The invited lectures, aimed at graduate stu-
dents in physics or mathematics, start with introductory material before
presentingmoreadvancedresults.Eachlectureisself-containedandcanbe
read independently of the rest.
The volume begins with the lectures on Spectral Geometry by Bruno
Iochum in which, strongly motivated by physics, the author presents the
fundamental concepts and results in the spectral approach to noncommu-
tative geometry. After the analytic preliminaries necessary to define the
spectral action for noncommutative spectral triples, computations are car-
ried out in some particular cases of interest in mathematical physics, such
asforRiemannianmanifolds(Einstein-Hilbertaction)andforthenoncom-
mutative torus.
The Atiyah-Singer index theorem and some of its generalizations are
the subject of the contribution by Maxim Braverman and Leonardo Cano,
based on the lectures given by the first author during the summer school.
The lectures start with a review of the mathematical tools required to
understand the significance of the classical Atiyah-Singer index theorem,
then provides a presentation of its equivariant counterpart and goes on
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vi Introduction
to a discussion of the recent developments aimed at generalizations of the
index theorem for transversally elliptic operators acting on non-compact
manifolds.
Thethirdlecture,byPaoloAluffi,ismotivatedbyanalgebro-geometric
interpretation of Feynman amplitudes in terms of periods. It starts with
an informal presentation of background in algebraic geometry on the
Grothendieck group of varieties and on characteristic classes, both viewed
as generalizations of the ordinary topological Euler characteristic. The last
part of the lecture reviews some recent work on Grothendieck classes and
Chern classes of graph hypersurfaces.
The fourth lecture byJorge Zanelli is an elementary introduction to
Chern-Simons forms aimed at understanding their role in physics and the
reason for their usefulness in gravity. The emphasis in this presentation is
on the construction of the action principles and on the geometric features
that make Chern-Simons forms particularly well suited for the physics of
geometric systems. The gauge invariance is reviewed, putting in the fore-
front the local Lorentz symmetry of space-time; the abundance of local
Lorentz-invariant gravitational theories is reduced by choosing the arbi-
trary coefficients in the Lagrangian in such a way that, in odd dimensions,
the gauge symmetry becomes enlarged.
Inthefifthlecture,MatildeMarcollicoversrecentapplicationstoparti-
cle physics of models inspired by noncommutative geometry. Complement-
ingtheformalmathematicalaspectsofthespectralactioncoveredbyBruno
Iochum’slectures,theauthordiscusseshowanactionfunctionalforgravity
coupledwithmatteron anordinaryspace-timecanbeseenas puregravity
on a noncommutative product geometry. Also, it is shown how the type of
noncommutative geometry involved determines the physical content of the
theory, i.e. the type of fermionic and bosonic fields present in the theory.
The lecture ends with an analysis of physical consequences of such particle
models which include the Higgs mass prediction.
ThesixthlecturebyA.P.Balachandranbeginswithareviewofthemo-
tivationfornoncommutativespacetimeswhicharisesfromquantumgravity.
AfteraphysicallymotivatedintroductiontoHopfalgebras,theconstruction
ofcovariantquantumfieldsonsuchspacetimesisdiscussed.Thisformalism
is then applied to cosmic mircowave background and to the prediction of
apparentnon-Paulitransitions.Thelectureendswithadiscussionoflimits
on the scale of noncommutativity, derived from experimental data.
Integrability, a way to test the so-called Maldacena conjecture, is the
theme of Matthias Staudacher’s Lecture. It first gives a presentation of the
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Introduction vii
Yang Baxter equations presented as an algebraic integrability condition by
meansoftheonedimensionalspinchainandtheBetheAnsatz.Theauthor
then discusses the AdS/CFT duality and its relation to integrability in a
N = 4 Supersymmetric Yang-Mills Theory, after which he describes some
predictions resulting from the theory.
MarianaGran˜aandHagenTriendl’scontribution,basedonthefirstau-
thor’s series of lectures in Villa de Leyva, provides an introduction to pure
spinors, generalized complex geometry and their role in flux compactifica-
tions in the context of string theory. After a short review of some useful
mathematical and physical tools in string theory and geometry, including
thetwistedintegrabilityofthealgebraicstructuretowhichthepurespinor
is attached, the authors give an interpretation of the manifolds arising in
flux compactification in string theory from the point of view of generalized
complex geometry.
Theinvitedlecturesarefollowedbyfourshortcommunicationsonawide
spectrum of topics on the borderline of mathematics and physics. In their
paper, Iv´an Contreras and Alberto Cattaneo give an overview of a novel
approach to symplectic groupoids in the context of Poisson sigma models
with boundary. In his contribution, Andr´es Angel reviews some classical
results on string topology of manifolds, and their recent extensions to the
theoryoforbifolds.PedroMorales-Almaz´an’scontribution,whichdiscusses
the Grothendieck ring class of banana and flower graphs, can be viewed as
an illustration of the lectures by P. Aluffi. The last communication in this
volume, by Rodrigo Vargas Le-Bert, studies the geometry underlying the
non-integrable and cyclic representations of a real Lie algebra.
Wehopethattheseself-containedandintroductorybutyetscientifically
very challenging lectures and short communications will serve as an incen-
tive for a broad reading audience of both mathematicians and physicists.
Wehopethattheywillmotivate,asmuchastheschoolitselfseemstohave
done,youngstudentstopursuewhatmightbetheirfirstacquaintancewith
some of the problems on the edge of mathematics and physics presented
here. We further hope that the more advanced reader will find some plea-
sure in reading about different outlooks on related topics and seeing how
thewell-knownmathematicaltoolsprovetobeveryusefulinsomeareasof
quantum field theory.
We are indebted to various organizations for their financial support for
this school. Let us first of all thank the Universidad de los Andes, which
was our main source of financial support in Colombia, and the Centre In-
ternational de Math´ematiques Pures et Appliqu´ees - CIMPA, for their sig-
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viii Introduction
nificantcontributiontothesuccessofthisschool.Otherorganizationssuch
as CLAF in Brazil, the Abdus Salam Internacional Centre for Theoretical
Physics and the International Mathematical Union, through the CDE pro-
gram,alsocontributedinasubstantialwaytothefinancialsupportneeded
for this event.
Special thanks to M´onica Castillo and Ana Catalina Salazar, who did a
great job for the practical organization of the school, the quality of which
was very much appreciated by participants and lecturers. We are also very
indebtedtoManuelTriana,LuzMalelyGuti´errezandMauricioMoralesfor
their help in various essential tasks needed for the successful development
of the school.
We would also like to thank the administrative staff at the Universi-
dad de los Andes, particularly Carl Langebaek, Vice-rector of Research,
Carlos Montenegro, Dean of the Faculty of Sciences, Alf Onshuus, Head
of the Mathematics Department, and Carlos A´vila, Head of the Physics
Department, for their constant encouragement and support.
Withoutthepeoplenamedhere,allofwhomhelpedintheorganization
in one way or another, before, during and after the school, this scientific
event would not have left such vivid memories in the lecturers’ and partic-
ipants’ minds. Last but not least, thanks to all the participants who gave
us all, lecturers and editors, the impulse to prepare this volume through
the enthusiasm they showed during the school, and thank you to all the
contributors and referees for their participation in the realization of these
proceedings.
The editors,
Alexander Cardona, Carolina Neira, Herna´n Ocampo, Sylvie Paycha∗
and Andr´es Reyes-Lega.
∗OnleavefromUniversityBlaisePascal,Clermont-Ferrand.
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ix
CONTENTS
Introduction v
Part A LECTURES 1
Spectral Geometry 3
B. Iochum
Index Theory for Non-compact G-manifolds 60
M. Braverman and L. Cano
GeneralizedEulerCharacteristics,GraphHypersurfaces,and
Feynman Periods 95
P. Aluffi
Gravitation Theory and Chern-Simons Forms 137
J. Zanelli
Noncommutative Geometry Models for Particle Physics 189
M. Marcolli
Noncommutative Spacetimes and Quantum Physics 224
A.P. Balachandran
Integrability and the AdS/CFT Correspondence 255
M. Staudacher
Compactifications of String Theory and Generalized Geometry 278
M. Gran˜a and H. Triendl